Venn Diagram

[goyalsau]

HI! Friends I have a problem in Set Theory Formula , Please Help me understand it


(reunion of A, B and C) = (total in A) + (total in B) + (total in C) - (total in exactly 2 groups) - 2(total in all 3 groups)

we have one more formula

# - Intersection ( I m not able to find the sign of intersection )

n ( A U B U C ) = n ( A ) + n ( B ) + n ( C ) - n ( A # B ) - n ( B # C ) - n ( C # A ) + n ( A # B # C )

I know both of the formula are same but i am not to figure out the difference How they are same,
Can any body please help me................

[goyalsau]

guys help

[Rahul@gurome]

HI! Friends I have a problem in Set Theory Formula , Please Help me understand it

(reunion of A, B and C) = (total in A) + (total in B) + (total in C) - (total in exactly 2 groups) - 2(total in all 3 groups)

we have one more formula

# - Intersection ( I m not able to find the sign of intersection )

n ( A U B U C ) = n ( A ) + n ( B ) + n ( C ) - n ( A # B ) - n ( B # C ) - n ( C # A ) + n ( A # B # C )

I know both of the formula are same but i am not to figure out the difference How they are same,
Can any body please help me................

Refer to the above image. If it is not clear then follow me...

Total in A - Blue
Total in B - Red
Total in C - Green

Total in (A ∩ B) = (Red + Blue) = Magenta
Total in (B ∩ C) = (Red + Green) = Yellow
Total in (A ∩ C) = (Blue + Green) = Cyan

Total in (A ∩ B ∩ C) = (Cyan + Magenta + Yellow) = (2*Red + 2*Blue + 2*Green) = White

Only in A and B = Magenta - White = (A ∩ B) - (A ∩ B ∩ C)
Only in B and C = Yellow - White = (B ∩ C) - (A ∩ B ∩ C)
Only in A and C = Cyan - White = (A ∩ C) - (A ∩ B ∩ C)

Therefore only in exactly two groups = Only in A and B + Only in B and C + Only in A and C = (A ∩ B) - (A ∩ B ∩ C) + (B ∩ C) - (A ∩ B ∩ C) + (A ∩ C) - (A ∩ B ∩ C) = (A ∩ B) + (B ∩ C) + (A ∩ C) - 3(A ∩ B ∩ C)

Only in A = Blue - Magenta - Cyan + White = A - (A ∩ B) - (A ∩ C) + (A ∩ B ∩ C)
Only in B = Red - Magenta - Yellow + White = B - (A ∩ B) - (B ∩ C) + (A ∩ B ∩ C)
Only in C = Green - Yellow - Cyan + White = C - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)

(White is added as Magenta, Cyan and Yellow also contains white in them, thus white is subtracted twice)

Now, 1st formula:

• (A U B U C) = (total in A) + (total in B) + (total in C) - (total in exactly 2 groups) - 2(total in all 3 groups)
  (A U B U C) = A + B + C - [(A ∩ B) + (B ∩ C) + (A ∩ C) - 3(A ∩ B ∩ C)] - 2*(A ∩ B ∩ C)
  (A U B U C) = A + B + C - (A ∩ B) - (B ∩ C) - (A ∩ C) + (A ∩ B ∩ C)]

Which is same as 2nd formula.

Hope it helps.

[Tani]

The second formula has an error. The last term should be to subtract twice the number that are in all three groups. Also, both formulas should include a term that adds in the number of items that are not in any of the groups. THat would give you the totals.

What you are doing is adding all the items, then subtracting those that are counted twice (i.e. in A and B, A and C, or B and C) and then subtracting two times the number of items that are in all three.

It may be easier to understand looking at the formula for combining two sets.

Total = A + B - both + neither

subtracting items in "both" compensates for the fact that they are counted twice. Adding "neither" picks up items that are not in any of the designated categories, but are included in the total.

Example = school has 100 students

45 take Spanish

63 take French

15 take neither

100 = 45 + 63 + 15 - B

B= number taking both French and Spanish. Here that would equal 23 students. You can show the same situation using a Venn diagram.

[goyalsau]

Refer to the above image.

Rahul, Thanks lot for the detailed explanation, But it would have been wonderful , IF i able to see the picture as well.

I am not able to open your picture, it says The file format is unsupported..

[Rahul@gurome]

@goyalsau: Updated the image. See if it helps.